An adaptive filter is a filter that self-adjusts its transfer function according to an optimizing algorithm. Most adaptive filters are digital filters that perform digital signal processing and adapt their performance based on the input signals. An adaptive filter has adaptive filter coefficients. Adaptive filters are especially suitable to those applications where some parameters of the desired signal processing operation are not known in advance. Such applications commonly employ adaptive filters that use feedback to refine the values of their adaptive filter coefficients and hence their frequency responses. In general, the adapting process involves the use of a cost function, which is a criterion for optimum performance of the adaptive filter, to feed an optimizing algorithm, which determines how to modify the adaptive filter coefficients to minimize the cost of the next iteration.
There are various implementations of the adaptive filters, including, for example, Least Mean Squares (LMS) filter and Recursive Least Squares (RLS) filter. A LMS filter uses least mean squares algorithms to find the adaptive filter coefficients that relate to producing the least mean squares of the error signal, which is the difference between the desired signal and the actual signal. It may be considered a stochastic gradient descent method in that the filter is adapted based on the error at the current time. A RLS filter uses recursive least squares algorithms to find the filter coefficients that relate to recursively producing the least squares (i.e., the minimum of the sum of the absolute squared) of the error signal. In some instances, the filter coefficients of an adaptive filter may be adjusted based at least in part on feedback signals to the adaptive filter.
Adaptive filters are routinely used in devices such as mobile telephones, digital cameras, camcorders, medical monitoring equipments, or tape drives for various purposes such as noise cancellation, signal prediction, adaptive feedback cancellation, or echo cancellation.